Finite Group Databasebeta

← #26112513

Subgroups — #26112513

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689318C211yesmaximal, characteristic
2C2×#15689201C211yesmaximal, characteristic, complemented
3#20244193C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic, complemented
5#20241271C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C47×#1802C211yesmaximal, characteristic, complemented
8C22×#10625334—21no—
9C22×#10625329—21no—
10C2×#13094021C2211yescharacteristic
11#15689318—22no—
12#15689318C414yescomplemented
13#15689201—22no—
14#15689201—22no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093944—21no—
18C2×#13093944—21no—
19C2×#13093944—21no—
20C2×C22⋊(C47⋊C8)—21no—
21C2×C4×C47⋊C8C2211yescharacteristic
22#15688866—22no—
23C2×C4×D4×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×#102×C47—21no—
26C47×#384C2211yescharacteristic
27C47×#340—21no—
28C22×D188—21no—
29C22×D188—21no—
30C22×D188—21no—
31C2×#10625334—24no—
32C2×#10625334—42no—
33C2×#10625334—42no—
34C2×#10625329—24no—
35C2×#10625329—42no—
36C2×#10625329—42no—
37C22×C47⋊C8—21no—
38C22×C47⋊C8—21no—
39C22×C47⋊C8D411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41#13094021—21no—
42#13094021—21no—
43#13094021—21no—
44#13094021C2×C412yes—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)C2×C412yes—
48C4⋊(C47⋊C8)D412yes—
49#13093944—24no—
50#13093944—42no—
51#13093944—42no—
52C22⋊(C47⋊C8)—24no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C22×D4×C47D411yescharacteristic
56C23×C4×C47—21no—
57C22×C8×C47—21no—
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47C2×C412yes—
61C4×D4×C47D412yes—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C22⋊C4×C47—21no—
67C2×C42×C47C2311yescharacteristic
68C4⋊C8×C47—21no—
69C4⋊C8×C47—21no—
70#102×C47—42no—
71C24×C47—21no—
72C23×D47—41no—
73C2×D4×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47—21no—
78C2×D4×C47C22⋊C412yes—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47#4111yescharacteristic
90C2×D188—22no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—41no—
102C2×D188—41no—
103C2×D188—41no—
104C2×D188—81no—
105C2×D188—81no—
106C2×C8×C47—41no—
107C2×C8×C47—41no—
108C2×C8×C47—41no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—81no—
115C4⋊C4×C47C22⋊C412yes—
116C22⋊C4×C47—22no—
117C22⋊C4×C47—22no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47#4111yescharacteristic
122#10625334—48no—
123#10625329—48no—
124C2×C47⋊C8—21no—
125C2×C47⋊C8—21no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8D812yescomplemented
136C23×C47—22no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47#12111yescharacteristic
144C22×D47—42no—
145C22×D47—81no—
146C22×D47—81no—
147C22×D47—81no—
148C22×D47—81no—
149C22×D47—81no—
150C22×D47—81no—
151D4×C47—22no—
152D4×C47—22no—
153D4×C47—41no—
154D4×C47—41no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—22no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—81no—
165C2×C4×C47—81no—
166C2×C4×C47—81no—
167C2×C4×C47C2×C22⋊C411yescharacteristic
168C2×C4×C47C4×D411yescharacteristic
169C2×C4×C47C4×D411yescharacteristic
170C2×C4×C47#12011yescharacteristic
171C2×C4×C47#12111yescharacteristic
172C2×C4×C47#12311yescharacteristic
173C2×C4×C47C2×D811yescharacteristic
174C2×C4×C47#13511yescharacteristic
175C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C47#13711yescharacteristic
181D188—42no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—82no—
186D188—82no—
187D188—81no—
188D188—81no—
189D188—81no—
190D188—81no—
191C8×C47—81no—
192C47⋊C8—22no—
193C47⋊C8—42no—
194C47⋊C8—42no—
195C47⋊C8—41no—
196#9229—471nomaximal, Sylow
197C22×C47—42no—
198C22×C47—42no—
199C22×C47—42no—
200C22×C47—81no—
201C22×C47—81no—
202C22×C47—81no—
203C22×C47—81no—
204C22×C47#38611yescharacteristic
205C22×C47#45911yescharacteristic
206C22×C47#46311yescharacteristic
207C22×C47#46811yescharacteristic
208C22×C47#46911yescharacteristic
209C22×C47#47111yescharacteristic
210C22×C47#47211yescharacteristic
211C2×D47—82no—
212C2×D47—82no—
213C2×D47—82no—
214C2×D47—161no—
215C2×D47—161no—
216C2×D47—161no—
217C2×D47—161no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—42no—
221C4×C47—161no—
222C4×C47#41711yescharacteristic
223C4×C47#41811yescharacteristic
224C4×C47C4×D811yescharacteristic
225C4×C47#44211yescharacteristic
226C4×C47#44711yescharacteristic
227C4×C47#44911yescharacteristic
228C4×C47#46811yescharacteristic
229C4×C47#46911yescharacteristic
230C2×#459—471no—
231C2×#436—471no—
232#1957—471no—
233#1802D4711yescharacteristic, complemented
234C2×#331—471no—
235#1296—471no—
236#1277—471no—
237C2×C47—82no—
238C2×C47#168511yescharacteristic
239C2×C47#168711yescharacteristic
240C2×C47#169711yescharacteristic
241C2×C47#170111yescharacteristic
242C2×C47#187911yescharacteristic
243C2×C47#189511yescharacteristic
244C2×C47#190711yescharacteristic
245D47—162no—
246C22×D8—941no—
247C22×M16—941no—
248C2×#127—471no—
249C2×C4×D4C2×D4711yescharacteristic
250#459—474no—
251#459—942no—
252#436—942no—
253#436—942no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8—471no—
256C2×C4⋊C8C2×D4711yescharacteristic
257C2×#102—21no—
258C2×#102—941no—
259C2×#102—941no—
260C2×#102—941no—
261C2×C22⋊C8—941no—
262C2×C4×C8—471no—
263#384C2×D4711yescharacteristic
264#340—21no—
265#331—942no—
266C47#922911yescharacteristic, complemented, Sylow
267C22×D4—941no—
268C22×D4—941no—
269C22×D4—941no—
270C22×D4C22⋊D4711yescharacteristic
271C23×C4—21no—
272C2×D8—944no—
273C2×D8—1882no—
274C2×D8—1882no—
275C2×M16—944no—
276C2×M16—1882no—
277C2×M16—1882no—
278C22×C8—21no—
279C22×C8—471no—
280C22×C8—471no—
281C22×C8—941no—
282C22×C8—941no—
283#127—472no—
284#127—941no—
285#127—941no—
286#127—941no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C4×D4712yes—
290C4×D4D18812yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C22⋊D4711yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D4711yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—472no—
300C4⋊C8—472no—
301C4⋊C8—941no—
302C4⋊C8—941no—
303#102—42no—
304#102—944no—
305#102—1882no—
306#102—1882no—
307C22⋊C8—944no—
308C4×C8—941no—
309C4×C8—941no—
310C24—21no—
311C24—1881no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—942no—
318C2×D4—942no—
319C2×D4—942no—
320C2×D4—3761no—
321C2×D4—3761no—
322C2×D4—1881no—
323C2×D4—1881no—
324C2×D4—1881no—
325C2×D4—1881no—
326C2×D4—1881no—
327C2×D4—1881no—
328C2×D4—1881no—
329C2×D4—1881no—
330C2×D4—1881no—
331C2×D4—1881no—
332C2×D4—1881no—
333C2×D4(C2×D47)⋊C412yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4D4×D4711yescharacteristic
343C22×C4#1062535211yescharacteristic
344C22×C4C2×C22⋊D4711yescharacteristic
345D8—1888no—
346M16—1888no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—472no—
351C2×C8—941no—
352C2×C8—941no—
353C2×C8—942no—
354C2×C8—941no—
355C2×C8—942no—
356C2×C8—941no—
357C2×C8—941no—
358C2×C8—1881no—
359C2×C8—1881no—
360C2×C8—1881no—
361C2×C8—1881no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—81no—
368C4⋊C4(C2×D47)⋊C412yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C2×C4×D4711yescharacteristic
372C42C2×D18811yescharacteristic
373C42D4×D4711yescharacteristic
374C42#1062535211yescharacteristic
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—3761no—
383C23—3761no—
384C23—3761no—
385C23—3761no—
386C23—3761no—
387C23—3761no—
388C23—1882no—
389C23#1309399611yescharacteristic
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—3762no—
395D4—3762no—
396D4—3761no—
397D4—3761no—
398D4—3761no—
399D4—3761no—
400D4—1882no—
401D4—1882no—
402D4—1882no—
403D4—1882no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4C2×(C2×D47)⋊C411yescharacteristic
417C2×C4(C22⋊D47)⋊C411yescharacteristic
418C2×C4(C22⋊D47)⋊C411yescharacteristic
419C2×C4#1309399411yescharacteristic
420C2×C4#1309399611yescharacteristic
421C2×C4#1309400811yescharacteristic
422C2×C4D8×D4711yescharacteristic
423C2×C4#1309404511yescharacteristic
424C2×C4#1309404811yescharacteristic
425C2×C4#1309404911yescharacteristic
426C2×C4#1309404911yescharacteristic
427C2×C4#1309405011yescharacteristic
428C2×C4#1309405111yescharacteristic
429C2×C4#1309405511yescharacteristic
430C8—81no—
431C8—942no—
432C8—1882no—
433C8—1881no—
434C8—1882no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—3762no—
443C22—3762no—
444C22—3762no—
445C22—7521no—
446C22—7521no—
447C22—7521no—
448C22—7521no—
449C22#1568900011yescharacteristic
450C22#1568931611yescharacteristic
451C22#1568934011yescharacteristic
452C22#1568937111yescharacteristic
453C22#1568937411yescharacteristic
454C22#1568938811yescharacteristic
455C22#1568939111yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—161no—
460C4#1568912211yescharacteristic
461C4#1568912711yescharacteristic
462C4#1568920311yescharacteristic
463C4#1568923211yescharacteristic
464C4#1568925811yescharacteristic
465C4#1568926911yescharacteristic
466C4#1568937111yescharacteristic
467C4#1568937411yescharacteristic
468C2—82no—
469C2—7522no—
470C2#2024285511yescharacteristic
471C2#2024286511yescharacteristic
472C2#2024292311yescharacteristic
473C2#2024295111yescharacteristic
474C2#2024373411yescharacteristic
475C2#2024382411yescharacteristic
476C2#2024389411yescharacteristic
subgroupG/H
centerC23#13093996
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1802C2
socleC23×C47#121
prime radicalC2×#15689318C2
cosocleC2×C42×C47C23
hypercenter#1802D47
hypocenterC47#9229
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#15689201C2semidirect
3. #20244193C2semidirect
4. C2×#15688866C2semidirect
5. #20241271C2semidirect
6. #20241222C2semidirect
7. C47×#1802C2semidirect
12. #15689318C4semidirect
135. C2×C47⋊C8D8semidirect
233. #1802D47semidirect
266. C47#9229semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156893182
C2×#156892012
#202441932
C2×#156888662
#202412712
#202412222
C47×#18022
#922947